International Journal of Chemical Sciences Research | Vol 15 Iss4

Regular Ternary Semigroups Jaya Lalitha G1, Sarala Y2 and Madhusudhana R3 1

Department of Mathematics, KL University, Guntur, Andhra Pradesh, India

2

Faculty of Mathematics, National Institute of Technology, Andhra Pradesh, India

3

Faculty of Mathematics, V.S.R & N.V.R College, Guntur, Andhra Pradesh, India

*

Corresponding author: Jaya Lalitha G, Department of Mathematics, KL University, Guntur, Andhra Pradesh, India,

Tel: 040 2354 2127; E-mail: jayalalitha.yerrapothu@gmail.com

Received: May 27, 2017; Accepted: August 29, 2017; Published: September 04, 2017

Abstract Intriguing properties of regular ternary semigroups and completely regular ternary semigroups were discussed in the article.

Keywords: Regular ternary semigroup; Completely regular ternary semigroup

Introduction Los [1] concentrated a few properties of ternary semigroups and demonstrated that each ternary semigroup can be installed in a semigroup. Sioson [2] concentrated ideal theory in ternary semigroups. He likewise presented the thought of regular ternary semigroups and characterized them by utilizing the thought of quasi ideals. Santiago [3] built up the theory of ternary semigroups and semiheaps. Dutta and Kar [4,5] presented and concentrated the thought of regular ternary semirings. Jayalalitha et al. [6] presented and learned about the filters in ternary semigroups. As of late, various mathematicians have taken a shot at ternary structures. In this paper, we concentrate some intriguing properties of regular ternary semigroups and completely regular ternary semigroups. Definition 1

An element x in a ternary semigroup T is said to be a regular if  an element a T  xax=x [2]. A ternary semigroup is said to be regular if all of its elements are regular.

Theorem 1 The following conditions in a ternary semigroup T are equivalent: (i) T is regular. (ii) For any right ideal R, lateral ideal M and left ideal L of T,

RML=R  M  L .

Citation: Jaya Lalitha G, Sarala Y, Madhusudhana R. Regular Ternary Semigroups. Int J Chem Sci. 2017;15(4):191

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(iii) For x, y, z  T ,

x

(iv) For x T ,

x

x

r

r

m

y

m

z l  x r y

xl  x r x

m

m

 z l.

 x l.

Proof

(i)  (ii) Suppose T is a regular ternary semigroup. Let R, M and L be a right ideal, a lateral ideal and a left ideal of T. Then clearly,

RML  R  M  L . Now for x  R  M  L , we have x=xax for some a  T . This implies that

x  xax  ( xax)(axa)( xax)  RML . Thus, we have

R  M  L  RML . So we find that RML= R  M  L .

Clearly, (ii)  (iii) and (iii)  (iv) . It remains to show that (iv)  (i) . Let x T . Clearly,

x x r  x

m

 xl  x

r

x

m

x l.

Then we have, x  ( xTT  nx)(TxT  TTxTT  nx)(TTx  nx)  xTx . So we find that

x xTa and hence there exists an elements a  T such that x=xax. This implies that x is regular and

hence T is regular.

We note that every left and right ideal of a regular ternary semigroup may not be a regular ternary semigroup. However, for a lateral ideal of a regular ternary semigroup, we have the following result:

Lemma Every lateral ideal of a regular ternary semigroup T is a regular ternary semigroup.

Proof Let L be a lateral ideal of regular ternary semigroup T. Then for each

x L

there exists

a T

such that x=xax. Now

x=xax=xaxax=x(axa)x=xpx where p = axa  L . This implies that L is a regular ternary semigroup.

Definition 2 An ideal A of a ternary semigroup T is said to be a regular ideal if lateral ideal

A  RML=R  M  L for any right ideal R  A ,

M  A and left ideal L  A .

Remark 1 From Definition 2, it follows that T is always a regular ideal and any ideal that contains a regular ideal is also a regular ideal. Now if for any right ideal R, lateral ideal M and left ideal L; RML contains a regular ideal, then

RML=R  M  L

Proposition A ternary semigroup T is a regular ternary semigroup if and only if 0 is a regular ideal of T. 2

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Proof Let P be the nuclear ideal of a ternary semigroup T. i.e., the intersection of all non-zero ideals of T, Pr is the intersection of all non-zero right ideals of T, Pm is the intersection of all non-zero lateral ideals of T and Pl is the intersection of all non-zero left ideals of T. Now if P= 0 , then clearly P=Pr=Pm=Pl.

Theorem 2 Let T be a ternary semigroup and P=Pr=Pm=Pl. Then T is a regular ternary semigroup if and only if P is a regular ideal of T.

Proof If P=Pr=Pm=Pl={0}, then proof follows from proposition. So we suppose that, P=Pr=Pm=Pl ≠ {0}. Let T be a regular ternary semigroup. Then from proposition, it follows that {0} is a regular ideal of T. Now, 0  P  Pr  Pm  Pl implies that P is a regular ideal of T, by using Remark 1. Conversely, let P be a regular ideal of T. Then and left ideal

P  RML=R  M  L for any right ideal R  P , lateral ideal M  P

L  P of T. Since PPP is a right ideal of T and P=Pr, we have P  Pr  PPP  RML .

Consequently,

P  RML=RML . So RML=R  M  L and hence from Theorem 2, it follows that T is a regular

ternary semigroup.

Corollary 1 Let T be a ternary semigroup and P=Pr=Pm=Pl. Then T is a regular ternary semigroup if and only if every ideal of T is regular.

Proof Suppose T is a regular ternary semigroup. Then from Theorem 2, it follows that P is a regular ideal of T. Now P=Pr=Pm=Pl implies that every non-zero ideal of T contains the regular ideal P of T. Consequently, by using Remark 1, we find that every ideal of T is regular.

Conversely, if every ideal of T is regular, then P is a regular ideal of T and hence from Theorem 2, it follows that T is a regular ternary semigroup.

Theorem 3 The following conditions in a ternary semigroup T are equivalent: (i) A is a regular ideal of T. (ii) For x, y, z  T , (iii) For x T ,

A x

A x

r

x

r

m

y

m

z l  A( x r  y

x l  A( x r  x

(iv) For each x  T \ A  A, x  a 

n i 1

xpi xqi x 

m

m

 z l ).

 x l).

n

xri si xui vi x for some a  A and pi , qi , ri , si , ui , vi  T .

i 1

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Proof

(i)  (ii) Suppose

A

is

a

regular

ideal

of

T.

We

note

for x, y, z  T ,

that

A  ( A  x r ), ( A  y m ), ( A  z l ). Now A  x  y r

m

 z l  ( A  x r )  (A  y m )  (A  z l )  A  ( A  x r ) (A  y m ) (A  z l ) (since A is regular).

 A  AAA  A y

 A x x

Again

r

r

y

y

m

z l  AA z l  x r AA  x r A z l  x

m

r

y

m

A x

r

y

m

z

l

z l.

m

z l x r y

m

A x

So we find that

A A y

y

r

m

m

 z l implies that A  x

z l  A( x r  y

r

y

z l  A x r  y

m

m

 z l.

 z l) .

m

(ii)  (iii) Put y=z=x in (ii) we get (iii). (iii)  (iv) We

first

note

A x

that

 A  ( xTT  nx)TT  A  xTTTT  nxTT  A  xTT A x

Similarly we have,

x r x

Now

m

m m

 A( A x

r

r

r

r

 A  x r  A  x r  T  T  A  x r TT

 A  xTT

 A  TxT  TTxTT and A  x

 x l  A x

 A( A x

r

r r

 A x  A x

 A x

r r

 A x

m m

m m

 A x

l

l

m m

l

l

 A x

l

l

 A  TTa.

l l

)

)

 A  ( A  xTT )( A  TxT  TTxTT )( A  TTx)  A  ( xTxTx  xTTxTTx)

x x r  x

Since,

m

n

n

i 1

i 1

 x

l

there

a A

exists

pi , qi , ri , si , ui , vi  T such

and

that

x  a  xpi xqi x   xri si xui vi x . (iv)  (i) Let R, M and L be any right, lateral and left ideal of T respectively such that R, M , L  A . Then clearly,

A  RML  R  M  L . n

n

i 1

i 1

Again,

let

x  a   xpi xqi x   xri si xui vi x for n

n

 xp xq x,  xr s xu v x  RML, i

i 1

i

i i

i i

x R  M  L . some

x  A  RML

a A and

Then

by and

hence

using

condition

(iv),

we

pi , qi , ri , si , ui , vi  T . R  M  L  A  RML.

have Since

Thus

i 1

A  RML = R  M  L . Consequently, A is a regular ideal.

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Theorem 4 Let A be a regular ideal of a ternary semigroup T. For any right ideal R, lateral ideal M and left ideal L of T, if then

RML A

R ML  A.

Proof Suppose for any right ideal R, lateral ideal M and left ideal L of T,

RML A , where A is a regular ideal of T. Then

A  ( A  R), ( A  M ), ( A  L) . Now R  M  L  ( A  R)  ( A  M )  ( A  L)

 A  (( A  R)( A  M )( A  L))

[Since

A

is

regular]

 A  AAA  AAL  AMA  AML  RAA  RAL  RMA  RML

 A. From Theorem 4, we have the following results:

Corollary 2 A regular and strongly irreducible ideal of a ternary semigroup T is a prime ideal of T.

Corollary 3 Every regular ideal of a ternary semigroup T is a semi prime ideal of T.

Theorem 5 A ternary semigroup T is regular if and only if every ideal of T is idempotent.

Proof Let T be a regular ternary semigroup and A be any ideal of T. Then A 3  AAA  TTA  A . Let exists

x  A . Then there

a  T such that x=xax=xaxax. Since A is an ideal and x  A , axa  A . Thus x = xax = xaxax  A3 .

Consequently, A  A3 and hence

A 3  AAA  A i.e., A is idempotent.

Conversely, suppose that every ideal of T is idempotent. Let P, Q and R be three ideals of T. Then PQR  PTT  P,

PQR  TQT  Q and PQR  TTR  R .

This

( P  Q  R) ( P  Q  R)( P  Q  R)  PQR .

Again,

( P  Q  R) ( P  Q  R)( P  Q  R)  P  Q  R .

that PQR  P  Q  R .

implies since Thus

( P  Q  R)

is

P  Q  R  PQR

an

ideal and

Also, of

T,

hence

P  Q  R  PQR . Therefore, by Theorem 2, T is a regular ternary semigroup. Theorem 6 A ternary semigroup T is left (resp. right) regular if and only if every left (resp. right) ideal of T is completely semiprime. 5

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Proof Let T be a left regular ternary semigroup and L be any left ideal of T. Suppose a  aaa  L for 3

regular, there exists an element

a  T . Since T is left

x  T such that a  xaa  x( xaa)a  xx(aaa) TTL  L. Thus L is completely

semiprime. Conversely, suppose that every left ideal of T is completely semiprime. Now for any a  T , by hypothesis,

Taa

is a left ideal of T. Then

Taa is a completely semiprime ideal of T. Now a3  aaa Taa . Since Taa is completely semiprime, it

follows that a  Taa . So there exists an element

x  T such that a=xaa. Consequently, a is left regular. Since a is

arbitrary, it follows that T is left regular.

Equivalently, we can prove the Theorem for right regularity.

Completely Regular Ternary Semigroup Definition 3 A pair (p, q) of elements in a ternary semigroup T is known as an idempotent pair if pq(pqx)=pqx and (xpq)pq=xpq for all

x  T [3]. Definition 4 Two idempotent pairs (p, q) and (r, s) of a ternary semigroup T are known as an equivalent, if pqx=rsx and xpq=xrs for all

x  T [3]. In notation we write (p, q) ~ (r, s).

Definition 5 An element x of a ternary semigroup T is said to be completely regular if

 an element a T  xax  x and the

idempotent pairs (a, x) and (x, a) are equivalent. If all the elements of T are completely regular, then T is called completely regular [3].

Definition 6 An element x of a ternary semigroup T is known as a left regular if

 an element a  T  axx=x

Definition 7 An element x of a ternary semigroup T is said to be right regular if

 an element a T  xxa  x

Theorem 7 A ternary semigroup T is completely regular then T is left and right regular. [i.e., x  x T  Tx for all 2

2

x  T ].

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Proof Suppose T is a completely regular ternary semigroup. Let

x  T . Then  an element a T  xax  x and the idempotent

pairs (x, a) and (a, x) are equivalent i.e., xab=axb and bxa=bax for all xax=axx and xaa=xax. This implies that

b  T . Now in particular, putting b=x we find that

x  xxT and x  Txx . Hence T is left and right regular.

Theorem 8 A ternary semigroup T is left and right regular then x  x Tx for all 2

2

x T .

Proof Suppose that T is both left and right regular. Let x  T . Then xpz=qxxpz=qxz for all

p, q  T  x  xxp and x=qxx. This implies that

z T .

Now x=xxp=x(xxp)p=x2(xpp)=x2(qxxpp)=x2(qxp)=x2q(qxx)p=x2 q2(xxp)=x2 q2x=x2 q2qxx=x2 q3x2  x Tx . Hence 2

2

x  x 2Tx 2 for all x  T . Theorem 9 If T is ternary semigroup x  x Tx for all x  T then T is completely regular. 2

2

Proof Suppose x  x Tx for all x  T . Then 2

Now

2

x  x2ax2  x( xax) x  xba ,

a  T  x  x2ax 2 where

b  xax  T .

This

implies

that

T

is

regular.

Also

xbc  x( xax)c  x 2 ax 2 c and bxc  ( xax) xc  x 2 ax 2 c for all c  T . This shows that the idempotent pairs (x, b) and (b, x) are equivalent. Consequently, T is a completely regular ternary semigroup.

Definition 8 A sub semigroup S of a ternary semigroup T is said to be a bi-ideal of T if STSTS  S.

Theorem 10 A ternary semigroup T is completely regular ternary semigroup if and only if every bi-ideal of T is completely semiprime.

Proof Let T is a completely regular ternary semigroup. Let P be any bi-ideal of T. Let regular, from Theorem 10, it follows that

p 3  P for p  T . Since T is completely

p  p 2Tp 2 . This implies that there exists x  T

such that

p  p 2 xp2  p( p 2 xp2 ) x( p 2 xp2 ) p  p3 ( xp 2 x) p( p 2 xp 2 ) xp3  p3 ( xp 2 x) p3 ( xp 2 x) p3  PTPTP  P .

This

shows that P is completely semiprime. 7

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Conversely, assume that every bi-ideal of T is completely semiprime. Since every left and right ideal of a ternary semigroup T is a bi-ideal of T, it follows that every left and right ideal of T is completely semiprime. Consequently, we have from Theorem 6 that T is both left and right regular. Now by using Theorem 9, we find that T is a completely regular ternary semigroup.

Theorem 11 If T is a completely regular ternary semigroup, then every bi-ideal of T is idempotent.

Proof Let T be a completely regular ternary semigroup and P be a bi-ideal of T. Clearly T is a completely regular ternary semigroup. Let

p  P. Then there exists x  T such that p=pxp. This implies that p  PTP and hence P  PTP. Also

PTP  PTPTP  P. Thus we find that P=PTP. Again, we have from Theorem 11 that p  p 2Tp 2  P 2TP 2 . This implies that

p  P 2TP 2  P( PTP) P  PPP  P . Hence P 3  P. Therefore every bi-ideal of P is idempotent.

Conclusion Ternary structures and their speculation, the purported n-ary structures bring certain expectations up in perspective of their conceivable applications in organic chemistry.

REFERENCES 1.

Los J. On the extending of models I. Fund Math. 1955;42:38-54.

2.

Sioson FM. Ideal theory in ternary semigroups. Math Japonica. 1965;10:63-84.

3.

Santiago ML. Some contributions to the study of ternary semigroups and semiheaps. Ph.D. Thesis. University of Madras 1983.

4.

Dutta TK, Kar S. On regular ternary semirings. Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific 2003;343-55.

5.

Dutta TK, Kar S. A note on regular ternary semirings. Kyungpook Mathematical Journal 46:357-65.

6.

Jayalalitha G, Sarala Y, Srinivasa Kumar B, et al. Filters in ternary semigroups. Int J Chem Sci. 2016;14:3190-4.

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Regular Ternary Semigroups

Los concentrated a few properties of ternary semigroups and demonstrated that each ternary semigroup can be installed in a semigroup. Sioson...

Regular Ternary Semigroups

Los concentrated a few properties of ternary semigroups and demonstrated that each ternary semigroup can be installed in a semigroup. Sioson...